Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-21T18:04:05.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Schemata: The Concept of Schema in the History of Logic

Published online by Cambridge University Press:  15 January 2014

John Corcoran
John Corcoran*
Affiliation:
University of Buffalo, Department of Philosophy, Buffalo, NY 14260-4150, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Schemata have played important roles in logic since Aristotle's Prior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski's 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano's second-order Induction Axiom is approximated by Herbrand's Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo's second-order Separation Axiom is approximated by Fraenkel's first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is a template-text or scheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with a side condition the template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, called instances of the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 12 , Issue 2 , June 2006 , pp. 219 - 240
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Audi, R. (editor), The Cambridge Dictionary of Philosophy, 2nd ed., Cambridge University Press, Cambridge, 1995/1999.Google Scholar
[2] Bernays, P., Axiomatic Set Theory, Dover, New York, 1958/1991.Google Scholar
[3] Boole, G., The Mathematical Analysis of Logic, Macmillan, Cambridge, 1847.Google Scholar
[4] Boolos, J., Burgess, G. and Jeffrey, R., Computability and Logic, Cambridge University Press, Cambridge, 2002.Google Scholar
[5] Carnap, R., Introduction to Symbolic Logic and its Applications, Dover, New York, 1954/1958.Google Scholar
[6] Church, A., Introduction to Mathematical Logic, Princeton University Press, Princeton, 1956.Google Scholar
[7] Cohen, M. and Nagel, E., Introduction to Logic, 2nd ed., Hackett, Indianapolis, 1993, 1st ed. Harcourt, Brace and World, New York, 1962, originally published as Book I of An Introduction to Logic and Scientific Method , Harcourt, Brace, and Company, New York, 1934.Google Scholar
[8] Corcoran, J., Conceptual structure of classical logic, Philosophy and Phenomenological Research, vol. 33 (1972), pp. 25–47.Google Scholar
[9] Corcoran, J., Scheme, In Audi [1], 1999.Google Scholar
[10] Corcoran, J., Schema, Stanford Encyclopedia of Philosophy, 2004.Google Scholar
[11] Corcoran, J., Meanings of word: type-occurrence-token, this Bulletin, vol. 11 (2005), p. 117.Google Scholar
[12] Corcoran, J., Frank, W., and Maloney, M., String theory, The Journal of Symbolic Logic, vol. 39 (1974), pp. 625–637.Google Scholar
[13] Davis, M., Computability and Unsolvability, 2nd ed., McGraw-Hill, Dover, New York, 1958/1980.Google Scholar
[14] Davis, M., The Undecidable, Raven Press, Hewlett, 1965.Google Scholar
[15] Encyclopedia Britannica, Chicago, London, Toronto, 1953.Google Scholar
[16] Feys, R. and Fitch, F. (editors), Dictionary of Symbols of Mathematical Logic, North-Holland Publishing Company, Amsterdam, 1969.Google Scholar
[17] Fraenkel, A., Introduction, [2], 1958/1991.Google Scholar
[18] Fraenkel, A., The notion “definite” and the independence of the axiom of choice, In Heijenoort [22], 1967, pp. 284–289.Google Scholar
[19] Frege, G., Philosophical and Mathematical Correspondence, Chicago University Press, Chicago, 1980.Google Scholar
[20] Gödel, K., The completeness of the axioms of the functional calculus of logic, In Heijenoort [22], 1967, pp. 582–591.Google Scholar
[21] Goldfarb, W., Deductive Logic, Hackett, Indianapolis, 2003.Google Scholar
[22] van Heijenoort, J. (editor), From Frege to Gödel, Harvard University Press, Cambridge, MA, 1967.Google Scholar
[23] Herbrand, J., Logical Writings, (Goldfarb, W., Goldfarb, Tr., and Heijenoort, van J., ecitors), Harvard University Press, Cambridge, MA, 1971.Google Scholar
[24] Hughes, R., Philosophical Companion to First-order Logic, Hackett, Indianapolis, 1993.Google Scholar
[25] Kleene, S., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936/1965), no. 5, pp. 727–742, also, in [14], pp. 237–254.Google Scholar
[26] Kleene, S., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943/1965), pp. 41–73, reprinted in [14], pp. 255–287.CrossRefGoogle Scholar
[27] Kleene, S., Introduction to Metamathematics, Van Nostrand, Princeton, 1952.Google Scholar
[28] Linsky, L., Semantics and the Philosophy of Language, University of Illinois Press, Urbana, 1952.Google Scholar
[29] Lyons, J., Semantics, Cambridge University Press, Cambridge, 1977, 2 Vols.Google Scholar
[30] Merriam-Webster's Collegiate Dictionary, Merriam-Webster, Springfield, MA, 2000.Google Scholar
[31] von Neumann, J., Zur Hilbertschen Beweistheorie, Mathematische Zeitschrift, vol. 26 (1927), pp. 1–46.Google Scholar
[32] New Shorter Oxford English Dictionary, Oxford University Press, Oxford, 1993.Google Scholar
[33] Ogden, C. K. and Richards, I. A., The Meaning of Meaning, 1923, reprint of the eighth edition, Harcourt, New York.Google Scholar
[34] Oxford English Dictionary, compact ed., Oxford University Press, Oxford, 1971.Google Scholar
[35] Peirce, C. S., Prolegomena to an apology for pragmaticism, Monist, vol. 16 (1906), pp. 492–546.Google Scholar
[36] Peirce, C. S., Collected papers of Charles Sanders Peirce, (Hartshorne, C. and Weiss, P., editors), Harvard University Press, Cambridge, 1933.Google Scholar
[37] Quine, W., On the logic of quantification, The Journal of Symbolic Logic, vol. 10 (1945), pp. 1–12.Google Scholar
[38] Quine, W., Mathematical Logic, Harvard University Press, Cambridge, MA, 1951.Google Scholar
[39] Quine, W., Philosophy of Logic, Harvard University Press, Cambridge, MA, 1970/1986.Google Scholar
[40] Rosenbloom, P. , Elements of Mathematical Logic, Dover, New York, 1950.Google Scholar
[41] Russell, B., Introduction to Mathematical Philosophy, George Allen and Unwin, London, 1919.Google Scholar
[42] Tarski, A., The concept of truth in the languages of the deductive sciences, Prace Towarzystwa Naukowego Warszawskiego, Wydzial III Nauk Matematyczno-Fizycznych, vol. 34 (1933), reprinted in [50], pp. 13–172; expanded English translation in [48], pp. 152–278.Google Scholar
[43] Tarski, A., Introduction to logic and to the methodology of deductive sciences, Dover, New York, 1941/1946/1995, Trans. Helmer, O..Google Scholar
[44] Tarski, A., Introduction to logic and to the methodology of deductive sciences, Oxford University Press, New York, 1941/1994, ed. with preface and biographical sketch of the author by J. Tarski.Google Scholar
[45] Tarski, A., The semantic conception of truth, Philosophy and Phenomenological Research, vol. 4 (1944), pp. 341–376, in [28], pp. 13–49.Google Scholar
[46] Tarski, A., Logic, Semantics, Metamathematics, papers from 1923 to 1938, Oxford University Press, Oxford, 1956.Google Scholar
[47] Tarski, A., Truth and proof, Scientific American, (1969), also, in [24], pp. 101–125.Google Scholar
[48] Tarski, A., Logic, Semantics, Metamathematics, papers from 1923 to 1938, 2nd ed., Hackett, Indianapolis, 1983, edited with introduction and analytic index by Corcoran, J..Google Scholar
[49] Vaught, R., Axiomatizability by a schema, The Journal of Symbolic Logic, vol. 32 (1967), pp. 473–479.Google Scholar
[50] Zygmunt, J. (editor), Alfred Tarski, Pisma Logiczno-Filozoficzne, 1 Prawda, Wydawnictwo Naukowe PWN, Warsaw, 1995.Google Scholar